Convolution & LTI systems worksheet with answers

Free printable Convolution & LTI systems worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
x[n]n−105152535…h[n]n−10−314213345

A system with the impulse response h[n] = {−3, 4, 1, 3} (n = 0…3) receives the step input x[n] = 5·u[n]. Find the output y[3].

h[n] = {−3, 4, 1, 3}x[n] = 5·u[n]
y[3] =
2.Easy
x[n]n−10−11−22334h[n]n−1011−32434

The sequences x[n] = {−1, −2, 3} and h[n] = {1, −3, 4} both start at n = 0. Find the largest sample ymax of y[n] = x[n] ∗ h[n].

x[n] = {−1, −2, 3}, n = 0…2h[n] = {1, −3, 4}, n = 0…2
ymax =
3.Medium
x(t)t−20h(t)t12

The signal x(t) is nonzero only between t = −2 s and t = 0 s, and h(t) only between t = 1 s and t = 2 s. Find the duration D of y(t) = x(t) ∗ h(t).

x: −2 … 0 sh: 1 … 2 s
D = s
4.Medium
x(t)t−11h(t)t37

The signal x(t) is nonzero only between t = −1 s and t = 1 s, and h(t) only between t = 3 s and t = 7 s. Find the time tstart at which y(t) = x(t) ∗ h(t) starts.

x: −1 … 1 sh: 3 … 7 s
tstart = s
5.Hard
5cos(5t)h(t) = 5e−5t​u(t)y(t)

The input x(t) = 5·cos(5t) is applied to a system with h(t) = 5·e−5tu(t). Find the phase φ of the steady-state output relative to the input, in degrees.

h(t) = 5e−5tu(t)x(t) = 5cos(5t)
φ = °
6.Hard
x(t)t (ms)2482−2

A triangle wave rises and falls linearly between −2 and +2 with period T = 4 ms. Find the amplitude c5 of its Fourier component at 1250 Hz.

±2T = 4 msf = 1250 Hz
c5 =

Answer key

  1. 1. y[3] = 25
  2. 2. ymax = 12
  3. 3. D = 3 s
  4. 4. tstart = 2 s
  5. 5. φ = −45 °
  6. 6. c5 = 0.06485

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